"Realise" is often a good translation for skjønne. Another translation for skjønne is something little kids say: "I don't get it!"
Å skjønne is related to that which is apparent (based on perception). Å forstå is related to that which is factual (can be proven).
The difference can be illustrated in the reading of a document in unfamiliar parlance (e.g., a high school student reading a legal brief), "I realise what it says, but I don't understand what it means," (Jeg skjønner det som står der, men jeg forstår ikke hva det betyr.)
Jeg skjønner ikke hva hun sier... Jeg måtte lytte den sakte versjonen.